The Stacks project

Lemma 25.9.1. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $L$ be a simplicial object of $\text{SR}(\mathcal{C}, X)$. Let $n \geq 0$. Consider the commutative diagram
\begin{equation} \label{hypercovering-equation-diagram} \xymatrix{ \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L)_{n + 1} \ar[r] \ar[d] & (\text{cosk}_ n \text{sk}_ n \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L))_{n + 1} \ar[d] \\ (L \times L)_{n + 1} \ar[r] & (\text{cosk}_ n \text{sk}_ n (L \times L))_{n + 1} } \end{equation}

coming from the morphism defined above. We can identify the terms in this diagram as follows, where $\partial \Delta [n + 1] = i_{n!}\text{sk}_ n \Delta [n + 1]$ is the $n$-skeleton of the $(n + 1)$-simplex:

\begin{eqnarray*} \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L)_{n + 1} & = & \mathop{\mathrm{Hom}}\nolimits (\Delta [1] \times \Delta [n + 1], L)_0 \\ (\text{cosk}_ n \text{sk}_ n \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L))_{n + 1} & = & \mathop{\mathrm{Hom}}\nolimits (\Delta [1] \times \partial \Delta [n + 1], L)_0 \\ (L \times L)_{n + 1} & = & \mathop{\mathrm{Hom}}\nolimits ( (\Delta [n + 1] \amalg \Delta [n + 1], L)_0 \\ (\text{cosk}_ n \text{sk}_ n (L \times L))_{n + 1} & = & \mathop{\mathrm{Hom}}\nolimits ( \partial \Delta [n + 1] \amalg \partial \Delta [n + 1], L)_0 \end{eqnarray*}

and the morphism between these objects of $\text{SR}(\mathcal{C}, X)$ come from the commutative diagram of simplicial sets
\begin{equation} \label{hypercovering-equation-dual-diagram} \xymatrix{ \Delta [1] \times \Delta [n + 1] & \Delta [1] \times \partial \Delta [n + 1] \ar[l] \\ \Delta [n + 1] \amalg \Delta [n + 1] \ar[u] & \partial \Delta [n + 1] \amalg \partial \Delta [n + 1] \ar[l] \ar[u] } \end{equation}

Moreover the fibre product of the bottom arrow and the right arrow in ( is equal to

\[ \mathop{\mathrm{Hom}}\nolimits (U, L)_0 \]

where $U \subset \Delta [1] \times \Delta [n + 1]$ is the smallest simplicial subset such that both $\Delta [n + 1] \amalg \Delta [n + 1]$ and $\Delta [1] \times \partial \Delta [n + 1]$ map into it.

Proof. The first and third equalities are Simplicial, Lemma 14.17.4. The second and fourth follow from the cited lemma combined with Simplicial, Lemma 14.21.11. The last assertion follows from the fact that $U$ is the push-out of the bottom and right arrow of the diagram (, via Simplicial, Lemma 14.17.5. To see that $U$ is equal to this push-out it suffices to see that the intersection of $\Delta [n + 1] \amalg \Delta [n + 1]$ and $\Delta [1] \times \partial \Delta [n + 1]$ in $\Delta [1] \times \Delta [n + 1]$ is equal to $\partial \Delta [n + 1] \amalg \partial \Delta [n + 1]$. This we leave to the reader. $\square$

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